For requesting, clarifying, and comparing definitions of mathematical terms.

learn more… | top users | synonyms

1
vote
0answers
98 views

Rapidly Decreasing Functions

Can someone explain the notion of a rapidly decreasing function? Namely, a function in the Schwartz space: $$\mathscr{S}(\mathbb{R}^n):= \{ f \in C^{\infty} (\mathbb{R}^n) : ||f||_{\alpha, \beta} ...
1
vote
0answers
120 views

Different definitions of uniform integrability

In different books and resources, I saw different definitions of uniformly integrable. For example, in some books the definition is like: Definition 1: $\{f_k\}\in L^1(E)$ is uniformly integrable ...
1
vote
0answers
69 views

definition of unitary operator

Wiki says " A bounded linear operator $U: H \to H$ on a Hilbert space $H$ is called a unitary operator if it satisfies $U^{*}U=UU^{*}=I$ , where $U^{*}$ is the adjoint of $U$, $I$ is the identity ...
1
vote
0answers
49 views

What's the name of this type operator?

If $H$, is a seperable Hilberspace, $E$ seperable Banach space. $(h_n)$ orthonormal basis of $H$. Let $T\in B(H,E)$. If we have the condition on $T$ that $$\sum_k \left\|Th_k\right\|^p <\infty,$$ ...
1
vote
0answers
46 views

What is real periodic function

I would like to know what is real periodic function. I understand what is periodic function, but I do not understand what is "real" periodic function.
1
vote
0answers
71 views

Is the proof of non-totality of a function $f$ trivial if $f$ is recursively defined only on a subset of $\Bbb N$?

I define a fuction $*$ in a recursive way on a subset of natural numbers (with subset I mean that is $*$ define in the second argument only for $\Bbb N-\{0,1\}$ in other words we have $*:\Bbb N ...
1
vote
0answers
42 views

Terminology: Name for general, non-differential equations over $\mathbb{R}^n$

Let $x \in \mathbb{R}^n$ be a variable, and $f_1, \ldots , f_n : \mathbb{R}^n \mapsto \mathbb{R}$ be a sequence of functions, each having a closed form expression. Assume that we want to solve the ...
1
vote
0answers
48 views

Motivation for defining a functions codomain

Why not always refer to a functions image, what is the point in specifying some super set of that image and then naming that super set the functions "codomain"? What does explicitly defining a set ...
1
vote
0answers
110 views

Meaning of $A^A$

Let $A$ be a matrix. I know that there is definition for $A^k$ power of $A$, and $e^A$ expontential of $A$. Is there any meaning of $A^A$?
1
vote
0answers
52 views

“Filters” of associative rings

Does there exist some good notion of "filters" for arbitrary associative rings with unity that would generalize filters of Boolean rings? For me, if $R$ is an associative ring with unity $1$, it ...
1
vote
0answers
54 views

Binary operation (english) terminology

Foreword: I have read R.H. Bruck's A Survey of binary systems, where the notion of halfoperation is given. A halfoperation $\ast$ differs from a (binary) operation since $a\ast b$ may not be defined ...
1
vote
0answers
15 views

acclivity of differential equation not defined

the differential equation y'*y=cos(x) has the acclivity of y'=cos(x)/y. Obviously the acclivity is not defined for points (x=r;y=0). r= real numbers Is there a way to give a short but logic argument ...
1
vote
0answers
36 views

What is a Complete Set of Weights of a Representation of a Lie Subalgebra?

In relation to Lie Group and Lie Algebra theory, I am studying about the weights of representations. I have come across the terminology "a complete string of weights" in my lecture course, but it is ...
1
vote
0answers
37 views

A question about the names of the components of an interval

Suppose you have the interval $[n,10n]$. How do you call the $n$ in this interval? I think the $n$ can be called the "independent variable of the interval" (since the interval $[n,10n]$ can be written ...
1
vote
0answers
14 views

Borel - Regular elements

In Borel's Linear Algebraic Groups (2ed) page 160 a regular element is defined in terms of its semisimple part, “thus $g$ is regular if and only if $g_s$ is regular.” A unipotent element $g$ has ...
1
vote
0answers
42 views

How is this function continuous?

The textbook stated that the following function whose domain is $\mathbb{R}$is continuous for every point in the domain: $g(x)=1, 0\le x\le1$ $=2, 2\le x \le3$, and it continues this patern. What ...
1
vote
0answers
35 views

Computing the differential (not the Jacobi-Matrix) independent of a basis choice

I am eager to learn more about the differential. I was told that it is a good practice to compute the differential independent of a choice of a basis using the following definition Definition: ...
1
vote
0answers
42 views

Indecomposable groups vs. indecomposable objects

An object $X$ in a category $\cal C$ with an initial object is called indecomposable if $X$ is not the initial object and $X$ is not isomorphic to a coproduct of two noninitial objects. A group $G$ ...
1
vote
0answers
32 views

Difficulty with terminology/standard definition for Jordan-normal form.

(Perhaps this is something rather for MOverflow, don't know) I've understood the concept of the Jordan-normal-form such that it is similar to the idea of diagonalization of a matrix, but can be ...
1
vote
0answers
121 views

Algebraic definition or construction of real numbers

Is there any algebraic definition or construction of real numbers ? If not, why ?
1
vote
0answers
59 views

Definition: foci of a quadric

How are the foci of a quadric defined? By a quadric I mean a set $$ Q = \left\{ x \in \mathbb R^n \mid x^T A x + 2 b^T x + c = 0 \right\}, $$ where $A\in\mathbb R^{n\times n}$ is symmetric and ...
1
vote
0answers
37 views

What is $(j,\epsilon)$-normality?

In looking at the concept of normality for real numbers I have come across the notion of $(j,\epsilon)$-normality, but cannot find a definition for this. could anyone explain what this term means?
1
vote
0answers
35 views

In a translation of a classic math book: is the term *normal* translated correctly? Is there a better term?

I am reading an English translation of the classic book by Gelfond & Linnik: Elementary Methods in the Analytic Theory of Numbers. Here is the definition of "normal" from the book (page 5 in my ...
1
vote
0answers
69 views

Understanding a.. weird definition

I came across the following definitons: Let $n$ be a positive integer and $\zeta_n$ be a primitive $n^{th}$ root of unity. For any prime $p\nmid n$, let the degree of the extension ...
1
vote
0answers
19 views

How do I state a reduction in cost?

I have developed an algorithm and am having a hard time stating its benefit versus a baseline. Suppose that the baseline cost of solving the problem is 1000 seconds. Now suppose that my algorithm ...
1
vote
0answers
87 views

$ |\sin (x) | \leq 1$ by it's definition

I was told that it can be shown that $|\sin(x)| \leq 1$ by its definition $\sin (z):= \frac{1}{2i} \bigl( (\exp(iz)-\exp(-iz)\bigr) $ I am aware that as soon as I choose $x \in \mathbb{R}$ and ...
1
vote
0answers
45 views

Subgraph without “holes”

does everyone know, if there already exists a definition of subgraphs, which do not contain a "hole"? EDITED: That means: I presuppose a planar embedding of a graph G and I want to find a connected ...
1
vote
0answers
89 views

$f: E \to \mathbb{R}^m$ is not continuous at $x_0$, show that $g:E \cup \lbrace x_0 \rbrace \to \mathbb{R}^m$ is continuous at $x_0$

Let $E \subset \mathbb{R}^k, f: E \to \mathbb{R^m} $ be a function. Let $x_0 \in \mathbb{R}^k$ such that $x_0 \notin E$ but note that $ \displaystyle \lim_{x \to x_0}_{x \in E}=y_0$ does exist. ...
1
vote
0answers
200 views

Difference between “marginal variance” and “true variance”?

[This question is not about solving a mathematical problem, but about definitions; please tell me if there is a better forum to ask it.] In a paper I am reading, the author writes: ...the sole ...
1
vote
0answers
80 views

About definition of topology

let be $m$ a function "$m: X \to \mathcal{P}(\mathcal{P}(X))$" (and I denote: $m(x) := m_x, \forall x \in X $), $m$ is topology on $X$ if: 2)$ \forall x \in X (\forall t \in m_x(x \in t)) $ 3)$ ...
1
vote
0answers
57 views

evaluating a meromorphic section of a line bundle at a point

Let $D$ be a Cartier divisor of a variety $X/K$ with associated line bundle $\mathcal{O}(D)$ and meromorphic section $s_D$. How do you define $s_D(P) \in K$ for $P \in X(K) \setminus ...
1
vote
0answers
375 views

Arc Length Parametrization

Professor was a little fuzzy on this topic, so I just wanted to make sure I have this definition correct: Given a function $\alpha : T \to C \mid t \in [a,b]$ , where $t$ is the parameter and $C$ is ...
1
vote
0answers
43 views

What's a algebraically isolated singularirty?

What's its formal definition and how does it differ from a "simple" isolated singularity? I'm thinking about integrable $1$-forms here, though the definition may be more general.
1
vote
0answers
71 views

How to Define an Infinite Anti-Diagonal Matrix

We can define an infinite diagonal matrix with some ease, and then say that a finite diagonal matrix is the top left sub-matrix of our infinite one. Can we define an infinite anti-diagonal matrix? It ...
1
vote
0answers
60 views

Is a single nontrivial convex set in a topological vector space enough to make it locally convex?

This is sort of a definition question. While a tvs is locally bounded if it contains a bounded neighborhood of the origin, a tvs is called locally convex if it contains a fundamental system of ...
1
vote
0answers
61 views

With infinite size, we can have $P \cdot M = M \cdot D $ (D diagonal) but where $M^{-1}$ does not exist. Can we say “P is diagonalizable”?

(I had this question in mind for longer time, but it is just triggered now by some comments at that recent question in mse) (Background) I was looking at properties of the Pascal-matrix: ...
1
vote
0answers
91 views

Which cut does the “minimum cut” refer to?

My course notes give the following definitions; could someone please verify that the last definition is non-standard? (I've spent all evening googling, and isn't "minimum cut" a concept related to cut ...
1
vote
0answers
112 views

How can I show using the $\epsilon-\delta$ definition of limit that $u(x,y_0)\to l$ as $x\to x_0?$

Let $u(x,y)\to l$ as $(x,y)\to z_0=(x_0,y_0).$ How can I show using the $\epsilon-\delta$ definition of limit that $u(x,y_0)\to l$ as $x\to x_0?$ Choose $\epsilon>0.$ Then ...
1
vote
0answers
38 views

What is the appropriate def. of $\sigma$-($\Sigma^1_1$) measurable.

I know that borel measurable means that the inverse image of a Borel set (or open set) is measurable. Edit: I am speaking of the sigma algebra generated by the analytic sets in a top. space.
1
vote
0answers
326 views

different kind of convergence in Real analysis

Now I am studying real analysis, I wonder if anyone can sum up all the relations between all kinds of convergence, convergence in measure, convergence a.e., convergence a.u., and convergence in ...
1
vote
0answers
94 views

Defining things in a non-recursive manner

In this question I asked, if it was possible to define certain functions without the use of the recursion theorem. The answer, among other things, indicated that it theoretically would be possible, ...
1
vote
0answers
24 views

Definition of Chromatic class digraph of a m-coloured digraph

I have to prove that if $D$ is a m-coloured digraph, then $K(D) = K\big(C(D)\big)$ Where $C(D)$ is the chromatic transitive closure of $D$, which is a multidigraph with the same nodes and arcs of $D$ ...
1
vote
0answers
172 views

How to prove the definition of arctangent by G. H. Hardy through integral?

From introduction to analysis,by Arthur P. Mattuck,problem 20-1. I am stuck in the sub-problem (d) of this problem,especially the magic number 2.5,please help,thanks. Problems 20-1 One way of ...
1
vote
0answers
98 views

How are the various numbers in the standard 2.2 gamma correction for RGB derived?

Here is the standard fwd Gamma 2.22 (1 / 0.45) correction formula: ...
1
vote
0answers
72 views

Definition of fragility

What does it mean for a solution to a system of differential equations to be fragile? A context for the term can be found here: This is taken from here in Mathematical Methods for Mechanics: A ...
1
vote
0answers
62 views

Are equidimensional ideals and unmixed ideals the same?

Zariski-Samuel define an unmixed/equidimensional ideal to be one whose associated primes have the same dimension. At other places I have seen definitions saying unmixed=all associated primes have same ...
1
vote
0answers
96 views

Definition of stereoprojection and Möbius maps

@WillieWong has kindly pointed out that there are 2 definitions of stereographic projection. One with the unit sphere placed on top of the plane, the other where the plane is at the equator of the ...
1
vote
0answers
173 views

Is the extra condition in this definition superfluous?

I am learning Differential Geometry and someone told me that the second condition of a definition provided in books follows from the first and is hence superfluous. I cannot dispute it, so convince me ...
0
votes
0answers
16 views

Notation: column/row projection function for matrix-like objects

If we have a $n$-tuple $\mathscr x$ $$\mathscr x := (x_i)_{i\in n}=(x_0,x_1,\ldots,x_{n-1})\in \prod_{i\in n}X_i$$ where $(X_i)_{i\in n}$ is an indexed family of sets and $x_i\in X_i$. We can ...
0
votes
0answers
17 views

Definition of convergence rate of random variables

What is the definition of rate of convergence of a sequence of random variables? i.e. what does it mean that the convergence rate of the sequence of random variable $X_1,X_2,\ldots X_n,\ldots$ to $X$ ...